Section 33.23 (0E8R): Separating points and tangent vectors

Lemma 33.23.1. Let $k$ be an algebraically closed field. Let $X$ be a proper $k$ -scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Let $V \subset H^0(X, \mathcal{L})$ be a $k$ -subvector space. If

for every pair of distinct closed points $x, y \in X$ there is a section $s \in V$ which vanishes at $x$ but not at $y$ , and
for every closed point $x \in X$ and nonzero tangent vector $\theta \in T_{X/k, x}$ there exists a section $s \in V$ which vanishes at $x$ but whose pullback by $\theta$ is nonzero,

then $\mathcal{L}$ is very ample and the canonical morphism $\varphi _{\mathcal{L}, V} : X \to \mathbf{P}(V)$ is a closed immersion.

Proof. Condition (1) implies in particular that the elements of $V$ generate $\mathcal{L}$ over $X$ . Hence we get a canonical morphism

$\varphi = \varphi _{\mathcal{L}, V} : X \longrightarrow \mathbf{P}(V)$

by Constructions, Example 27.21.2. The morphism $\varphi$ is proper by Morphisms, Lemma 29.41.7. By (1) the map $\varphi$ is injective on closed points (computation omitted). In particular, the fibre over any closed point of $\mathbf{P}(V)$ is a singleton (small detail omitted). Thus we see that $\varphi$ is finite, for example use Cohomology of Schemes, Lemma 30.21.2. To finish the proof it suffices to show that the map

$\varphi ^\sharp : \mathcal{O}_{\mathbf{P}(V)} \longrightarrow \varphi _*\mathcal{O}_ X$

is surjective. This we may check on stalks at closed points. Let $x \in X$ be a closed point with image the closed point $p = \varphi (x) \in \mathbf{P}(V)$ . Since $\varphi ^{-1}(\{ p\} ) = \{ x\}$ by (1) and since $\varphi$ is proper (hence closed), we see that $\varphi ^{-1}(U)$ runs through a fundamental system of open neighbourhoods of $x$ as $U$ runs through a fundamental system of open neighbourhoods of $p$ . We conclude that on stalks at $p$ we obtain the map

$\varphi ^\sharp _ x : \mathcal{O}_{\mathbf{P}(V), p} \longrightarrow \mathcal{O}_{X, x}$

In particular, $\mathcal{O}_{X, x}$ is a finite $\mathcal{O}_{\mathbf{P}(V), p}$ -module. Moreover, the residue fields of $x$ and $p$ are equal to $k$ (as $k$ is algebraically closed – use the Hilbert Nullstellensatz). Finally, condition (2) implies that the map

$T_{X/k, x} \longrightarrow T_{\mathbf{P}(V)/k, p}$

is injective since any nonzero $\theta$ in the kernel of this map couldn't possibly satisfy the conclusion of (2). In terms of the map of local rings above this means that

$\mathfrak m_ p/\mathfrak m_ p^2 \longrightarrow \mathfrak m_ x/\mathfrak m_ x^2$

is surjective, see Lemma 33.16.5. Now the proof is finished by applying Algebra, Lemma 10.20.3. $\square$

Lemma 33.23.2. Let $k$ be an algebraically closed field. Let $X$ be a proper $k$ -scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Suppose that for every closed subscheme $Z \subset X$ of dimension $0$ and degree $2$ over $k$ the map

$H^0(X, \mathcal{L}) \longrightarrow H^0(Z, \mathcal{L}|_ Z)$

is surjective. Then $\mathcal{L}$ is very ample on $X$ over $k$ .

Proof. This is a reformulation of Lemma 33.23.1. Namely, given distinct closed points $x, y \in X$ taking $Z = x \cup y$ (viewed as closed subscheme) we get condition (1) of the lemma. And given a nonzero tangent vector $\theta \in T_{X/k, x}$ the morphism $\theta : \mathop{\mathrm{Spec}}(k[\epsilon ]) \to X$ is a closed immersion. Setting $Z = \mathop{\mathrm{Im}}(\theta )$ we obtain condition (2) of the lemma. $\square$

The Stacks project

33.23 Separating points and tangent vectors

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